QN 9 From past experience a statistics professor knows that

Q.N. 9) From past experience, a statistics professor knows that 85% of his students who do the test review problems pass the test, and of those who do not do the review problems, 90% fail. Before he starts to grade the latest test, he believes that 95% of his students did the review problems. The first randomly selected test he grades receives a failing grade. Find the conditional probability that the student did the review. Q.N. 10) If A and B are independent events, prove that A^c and B^c are independent. Note that A^c and B^c are the compliments of A and B respectively.

Solution

9.

let A be the event of the student failing the test.

let B be the event of the student did the review.

=>

P(AnB) = 0.95*0.15 =0.1425

P(B) = 0.95*0.15 + 0.05* 0.90 = 0.1875

=>

the required probability = P(A|B) = 0.1425/0.1875 = 0.76

10)

P(AnB) = P(A)P(B)

P(A\' n B\') = 1-P(AuB) = 1- [P(A) + P(B) -P(AnB)]

= 1-P(A)-P(B) + P(AnB)

= 1-P(A)-P(B) + P(A)P(B)

= (1-P(A))(1-P(B))

= P(A\')P(B\')

=>

P(A\'nB\') = P(A\')P(B\')

=>

A\', B\' are independent events

thus proved